CN103954948A - Fuzzy hypothesis test using fuzzy data and confidence interval in radar detection standard - Google Patents

Abstract

The invention provides a fuzzy hypothesis test algorithm when hypothesis and data are both fuzzy values in a judgment standard of radar detection signals. Specifically speaking, a confidence interval is firstly provided to given fuzzy data and then a fuzzy method is used to calculate a related fuzzy test statistical quantity. The fuzzy hypothesis test algorithm is a soft judgment method which can be represented by a membership function. The method is capable of overcoming a plurality of disadvantages of the prior judgment methods, such as that the judgment result is sensitive to the mean value x<-> of the fuzzy data and that only one judgment result is obtained and the like.

Description

In radar detecting standard, use the Fuzzy Hypothesis Testing between fuzzy data and trusted domain

Technical field

The present invention is to provide in the detecting standard of radar signal a kind of Fuzzy Hypothesis Testing algorithm in the time supposing and data are all fuzzy value.Specifically, first for given fuzzy data provides between a trusted domain, then calculate relevant Fuzzy Test statistic with blur method, it is a kind of soft method for detecting that can represent with subordinate function.The method can overcome some weakness in other existing decision method, as the average of court verdict to fuzzy data sensitivity, and only have a court verdict etc.

Background technology

In traditional radar detection system, judge that for the signal receiving it is target or noise, employing be a kind of method of hard decision, this signal only has two kinds of possibilities: be not that target is exactly noise.This judgement mode is applied in some false judgment of the middle inevitably generation of radar (as tracking radar) of some kind.Although have a lot of scholars from mathematical angle at present, from Fuzzy Hypothesis Testing being studied between fuzzy data or fuzzy hypothesis or fuzzy trusted domain etc., but the concrete grammar for radar signal fuzzy judgment in fuzzy hypothesis and the simultaneous situation of fuzzy data, few people study.

This patent has proposed a kind ofly not only to have considered fuzzy hypothesis but also considered a kind of fuzzy decision method in the situation of fuzzy data simultaneously.This court verdict is to provide court verdict according to the concrete radar type adopting in the mode of a certain decision-making degree, and it can be according to the fuzzy hypothesis that a certain degree is accepted or refusal is predetermined.

Summary of the invention

The object of the present invention is to provide a kind of given assumed condition of working as, the radar detecting decision method when data that radar receives and error are all fuzzy data.The method can allow deviser can have four kinds of different types of selections in the time of design Radar Receiver System.

Make X1 ..., X _nbe the energy signal of the independent sample with same distribution that receives from radar receiver, its distribution is the known σ of unknown μ ²standard probability density function.Suppose that the random sample of observing is fuzzy data instead of clear data, we carry out Fuzzy Hypothesis Testing by I type error, type-I error=β here _oalso be fuzzy data.

Suppose: $H_o:\mathrm{\&mu;}={\mathrm{\&mu;}}_o^{*}\mathrm{Microwatt}$ (noise or driftlessness)

$H_1:\mathrm{\&mu;}={\mathrm{\&mu;}}_o^{*}\mathrm{Microwatt}$ (having target)

Now, $\stackrel{\&OverBar;}{x}=T(a_1,{\stackrel{\&OverBar;}{x}}_o,a_3)=[{\stackrel{\&OverBar;}{x}}_L\left(\mathrm{\&alpha;}\right),{\stackrel{\&OverBar;}{x}}_U\left(\mathrm{\&alpha;}\right)]=[a_1+({\stackrel{\&OverBar;}{x}}_o-a_1)\mathrm{\&alpha;},a_3-(a_3-{\stackrel{\&OverBar;}{\text{x}}}_o)\mathrm{\&alpha;}],$ ? value is determined by fuzzy data.Here, $T(a,b,c)\left(x\right)=\left\{\begin{array}{c}\frac{x-a}{b-a},a<x\&le;b\\ \frac{x-c}{b-c},b<x\&le;c\\ 0,\mathrm{otherwise}\end{array}\right.,$ As shown in Figure 1.

${\mathrm{\&mu;}}_o^{*}=T(a_1^{\&prime;},{\mathrm{\&theta;}}_o,a_3^{\&prime;})=[a_1^{\&prime;}+({\mathrm{\&theta;}}_o-a_1^{\&prime;})\mathrm{\&alpha;},a_3^{\&prime;}-(a_3^{\&prime;}-{\mathrm{\&theta;}}_o)\mathrm{\&alpha;}],$ value is determined by fuzzy hypothesis.β＝T(a″ ₁，β _o，a″ ₃)。

Specific implementation step of the present invention is as follows:

1), according to computational data between fuzzy data and fuzzy trusted domain

${\stackrel{\&OverBar;}{x}}_1\left[\mathrm{\&alpha;}\right]=[{\stackrel{\&OverBar;}{x}}_L\left(\mathrm{\&alpha;}\right)-Z_{1-\frac{\mathrm{\&alpha;}}{2}}(\mathrm{\&sigma;}/\sqrt{n}),{\stackrel{\&OverBar;}{x}}_L\left(\mathrm{\&alpha;}\right)+Z_{1-\frac{\mathrm{\&alpha;}}{2}}(\mathrm{\&sigma;}/\sqrt{n})]$

${\stackrel{\&OverBar;}{x}}_2\left[\mathrm{\&alpha;}\right]=[{\stackrel{\&OverBar;}{x}}_U\left(\mathrm{\&alpha;}\right)-Z_{1-\frac{\mathrm{\&alpha;}}{2}}(\mathrm{\&sigma;}/\sqrt{n}),{\stackrel{\&OverBar;}{x}}_U\left(\mathrm{\&alpha;}\right)+Z_{1-\frac{\mathrm{\&alpha;}}{2}}(\mathrm{\&sigma;}/\sqrt{n})]$

Here: ${\stackrel{\&OverBar;}{x}}_L\left(\mathrm{\&alpha;}\right)=a_1+({\stackrel{\&OverBar;}{x}}_o-a_1)\mathrm{\&alpha;},{\stackrel{\&OverBar;}{x}}_U\left(\mathrm{\&alpha;}\right)=a_3-(a_3-{\stackrel{-}{x}}_o)\mathrm{\&alpha;}$

2), calculate owing to supposing the fuzzy of generation:

${\mathrm{\&mu;}}_o^{*}=T(a_1^{\&prime;},{\mathrm{\&theta;}}_o,a_3^{\&prime;})=[a_1^{\&prime;}+({\mathrm{\&theta;}}_o-a_1^{\&prime;})\mathrm{\&alpha;},a_3^{\&prime;}-(a_3^{\&prime;}-{\mathrm{\&theta;}}_o)\mathrm{\&alpha;}]$

3), calculate Fuzzy Test statistic:

${\tilde{Z}}_1\left[\mathrm{\&alpha;}\right]=[\frac{{\stackrel{\&OverBar;}{x}}_L\left(\mathrm{\&alpha;}\right)-Z_{1-\frac{\mathrm{\&alpha;}}{2}}(\mathrm{\&sigma;}/\sqrt{n})-a_3^{\&prime;}+(a_3^{\&prime;}-{\mathrm{\&theta;}}_o)\mathrm{\&alpha;}}{\mathrm{\&sigma;}/\sqrt{n}},\frac{{\stackrel{\&OverBar;}{x}}_L\left(\mathrm{\&alpha;}\right)+Z_{1-\frac{\mathrm{\&alpha;}}{2}}(\mathrm{\&sigma;}/\sqrt{n})-a_1^{\&prime;}-({\mathrm{\&theta;}}_o-a_1^{\&prime;})\mathrm{\&alpha;}}{\mathrm{\&sigma;}/\sqrt{n}}]$

${\tilde{Z}}_2\left[\mathrm{\&alpha;}\right]=[\frac{{\stackrel{\&OverBar;}{x}}_U\left(\mathrm{\&alpha;}\right)-Z_{1-\frac{\mathrm{\&alpha;}}{2}}(\mathrm{\&sigma;}/\sqrt{n})-a_3^{\&prime;}+(a_3^{\&prime;}-{\mathrm{\&theta;}}_o)\mathrm{\&alpha;}}{\mathrm{\&sigma;}/\sqrt{n}},\frac{{\stackrel{\&OverBar;}{x}}_U\left(\mathrm{\&alpha;}\right)+Z_{1-\frac{\mathrm{\&alpha;}}{2}}(\mathrm{\&sigma;}/\sqrt{n})-a_1^{\&prime;}-({\mathrm{\&theta;}}_o-a_1^{\&prime;})\mathrm{\&alpha;}}{\mathrm{\&sigma;}/\sqrt{n}}]$

4), draw subordinate function, as shown in Figure 2.

5), calculate:

① $\tilde{Z}\left[\mathrm{\&alpha;}\right]={\tilde{Z}}_1\left[\mathrm{\&alpha;}\right]\&cup;{\tilde{Z}}_2\left[\mathrm{\&alpha;}\right],$ In the time that ambiguity reaches maximum;

② $\tilde{Z}\left[\mathrm{\&alpha;}\right]={\tilde{Z}}_1\left[\mathrm{\&alpha;}\right]\&cap;{\tilde{Z}}_2\left[\mathrm{\&alpha;}\right],$ When ambiguity reaches hour;

3. work as H _orefusal value minimum (for right-sided test);

4. work as H _orefusal value maximum (for right-sided test).

6), suppose and the subordinate function of I type error is Z _β[α]=[h ₁(α), h ₂(α)].

7), calculate

$Z\left[\mathrm{\&alpha;}\right]>Z_{\mathrm{\&beta;}}\left[\mathrm{\&alpha;}\right]=\frac{k_2\left(\mathrm{\&alpha;}\right)-h_1\left(\mathrm{\&alpha;}\right)}{(h_2\left(\mathrm{\&alpha;}\right)-h_1\left(\mathrm{\&alpha;}\right))+(k_2\left(\mathrm{\&alpha;}\right)-k_1\left(\mathrm{\&alpha;}\right))}$

$\tilde{Z}>{\tilde{Z}}_{\mathrm{\&beta;}}=\frac{\underset{\mathrm{\&alpha;}=0}{\overset{\mathrm{\&alpha;}<1}{\mathrm{\&Sigma;}}}\left((Z\left[\mathrm{\&alpha;}\right]>Z_{\mathrm{\&beta;}}\left[\mathrm{\&alpha;}\right]\&CenterDot;(k_2\left(\mathrm{\&alpha;}\right)-k_1\left(\mathrm{\&alpha;}\right))\&CenterDot;(h_2\left(\mathrm{\&alpha;}\right)-h_1\left(\mathrm{\&alpha;}\right)))\right)}{\underset{\mathrm{\&alpha;}=0}{\overset{\mathrm{\&alpha;}<1}{\mathrm{\&Sigma;}}}((k_2\left(\mathrm{\&alpha;}\right)-k_1\left(\mathrm{\&alpha;}\right))\&CenterDot;(h_2\left(\mathrm{\&alpha;}\right)-h_1\left(\mathrm{\&alpha;}\right)))}$

Here, for any alpha levels, $k_1\left(\mathrm{\&alpha;}\right)=\underset{x}{\min }Z\left[\mathrm{\&alpha;}\right],k_2\left(\mathrm{\&alpha;}\right)=\underset{x}{\max }Z\left[\mathrm{\&alpha;}\right],h_1\left(\mathrm{\&alpha;}\right)=\underset{x}{\min }{Z}_{\mathrm{\&beta;}}\left[\mathrm{\&alpha;}\right],$ $h_2\left(\mathrm{\&alpha;}\right)=\underset{x}{\max }{Z}_{\mathrm{\&beta;}}\left[\mathrm{\&alpha;}\right].$

8) if $H_o=\tilde{Z}\text{>}{\tilde{Z}}_{\mathrm{\&beta;}},$ Refusal hypothesis.

There are following four kinds of results in this algorithm:

1. work as time, result is the same with the algorithm that M.Arefi and S.M.Taheri proposes, and exists maximum blur level.

2. work as time, result exists least confusion degree.If with be symmetrical, this result is 1. identical with situation.Otherwise this result depends on selected radar type.

3. work as time, that draw is refusal H _ominimum value.

4. work as time, that draw is refusal H _omaximal value.

Brief description of the drawings

The subordinate function of the fuzzy hypothesis of Fig. 1

Fig. 2 subordinate function

Fig. 3 curve represents

Fig. 4 curve represents

Fig. 5 curve represents

Fig. 6 curve represents

Embodiment

First definition: suppose H _otable driftlessness, supposes H ₁table has target;

" false alarm rate "=P _fa=β=I type error probability=P (refusal H _o| H _obe true);

" false dismissed rate "=λ=II type error probability=P (accepts H _o| H ₁be true);

" verification and measurement ratio "=1-λ;

Dualism hypothesis: $H_o:\mathrm{\&mu;}={\widetilde{\mathrm{\&mu;}}}_o$ (noise)

$H_1:\mathrm{\&mu;}>{\widetilde{\mathrm{\&mu;}}}_o$ (target)

Make x ₁..., x ₁₀₀for 100 of receiving from radar receiving trap have the power of the independently random signal that equal probabilities distributes, it is distributed as normal distribution, wherein, and μ the unknown, σ ²=9.Utilize method of the present invention I type error P _fa=β _o=0.005 carries out test of hypothesis.

$H_o:\mathrm{\&mu;}={\mathrm{\&mu;}}_o^{*}\mathrm{Microwatt}$ (noise or driftlessness)

$H_1:\mathrm{\&mu;}={\mathrm{\&mu;}}_o^{*}\mathrm{Microwatt}$ (having target)

Here, $\stackrel{\&OverBar;}{x}=T(a_1,{\stackrel{\&OverBar;}{x}}_o,a_3)=T\left(\mathrm{1.6,1.8,2}\right)$

${\mathrm{\&mu;}}_o^{*}=T(a_1^{\&prime;},{\mathrm{\&theta;}}_o,a_3^{\&prime;})=T\left(\mathrm{0,1,2}\right)$

β＝T(a″ ₁，β _o，a″ ₃)＝T(0.003，0..5，0.007)

The 1st) step:

${\stackrel{\&OverBar;}{x}}_L\left(\mathrm{\&alpha;}\right)=1.6+0.2\mathrm{\&alpha;}$ With ${\stackrel{\&OverBar;}{x}}_U\left(\mathrm{\&alpha;}\right)=2-0.2\mathrm{\&alpha;}$

${\stackrel{\&OverBar;}{x}}_1\left[\mathrm{\&alpha;}\right]=[1.6+0.2\mathrm{\&alpha;}-Z_{1-\frac{\mathrm{\&alpha;}}{2}}(3/10),1.6+0.2\mathrm{\&alpha;}+Z_{1-\frac{\mathrm{\&alpha;}}{2}}(3/10)],$ As shown in Figure 3.

${\stackrel{\&OverBar;}{x}}_2\left[\mathrm{\&alpha;}\right]=[2+0.2\mathrm{\&alpha;}-Z_{1-\frac{\mathrm{\&alpha;}}{2}}(3/10),2+0.2\mathrm{\&alpha;}+Z_{1-\frac{\mathrm{\&alpha;}}{2}}(3/10)],$ As shown in Figure 4.

The 2nd) step: calculate ${\mathrm{\&mu;}}_o^{*}=T\left(\mathrm{0,1,2}\right)=[\mathrm{\&alpha;},2-\mathrm{\&alpha;}]$

The 3rd) step: calculate ${\tilde{Z}}_1\left[\mathrm{\&alpha;}\right]=[\frac{1.6+0.2\mathrm{\&alpha;}-Z_{1-\frac{\mathrm{\&alpha;}}{2}}(3/10)-(2-\mathrm{\&alpha;})}{3/10},\frac{1.6+0.2\mathrm{\&alpha;}+Z_{1-\frac{\mathrm{\&alpha;}}{2}}(3/10)-\mathrm{\&alpha;}}{3/10}],$

${\tilde{Z}}_2\left[\mathrm{\&alpha;}\right]=[\frac{2-0.2\mathrm{\&alpha;}-Z_{1-\frac{\mathrm{\&alpha;}}{2}}(3/10)-(2-\mathrm{\&alpha;})}{3/10},\frac{2-0.2\mathrm{\&alpha;}+Z_{1-\frac{\mathrm{\&alpha;}}{2}}(3/10)-\mathrm{\&alpha;}}{3/10}]$

The 4th) step: obtain subordinate function, as Fig. 5, shown in Fig. 6.

The 5th) step: calculate 1. $\tilde{Z}\left[\mathrm{\&alpha;}\right]={\tilde{Z}}_1\left[\mathrm{\&alpha;}\right]\&cup;{\tilde{Z}}_2\left[\mathrm{\&alpha;}\right]$

② $\tilde{Z}\left[\mathrm{\&alpha;}\right]={\tilde{Z}}_1\left[\mathrm{\&alpha;}\right]\&cap;{\tilde{Z}}_2\left[\mathrm{\&alpha;}\right]$

③ $\tilde{Z}\left[\mathrm{\&alpha;}\right]={\tilde{Z}}_1\left[\mathrm{\&alpha;}\right]$

④ $\tilde{Z}\left[\mathrm{\&alpha;}\right]={\tilde{Z}}_2\left[\mathrm{\&alpha;}\right]$

The 6th) step: if 1. h _o=0.50617, refusal hypothesis H _o

If 2. h _o=0.5074, refusal hypothesis H _o

If 3. h _o=0.4600, refusal hypothesis H _o

If 4. h _o=0.5534, refusal hypothesis H _o

The method compares with three kinds of now conventional methods, and comparative result as shown in Table.

Can find out that by above comparative result the inventive method has the superiority that other existing classical way does not possess.

Claims (1)

1. a Fuzzy Hypothesis Testing method for the radar detection signal in the time supposing and data are all fuzzy value, described method comprises following concrete steps:

1), according to computational data between fuzzy data and fuzzy trusted domain

${\stackrel{\&OverBar;}{x}}_1\left[\mathrm{\&alpha;}\right]=[{\stackrel{\&OverBar;}{x}}_L\left(\mathrm{\&alpha;}\right)-Z_{1-\frac{\mathrm{\&alpha;}}{2}}(\mathrm{\&sigma;}/\sqrt{n}),{\stackrel{\&OverBar;}{x}}_L\left(\mathrm{\&alpha;}\right)+Z_{1-\frac{\mathrm{\&alpha;}}{2}}(\mathrm{\&sigma;}/\sqrt{n})]$

${\stackrel{\&OverBar;}{x}}_2\left[\mathrm{\&alpha;}\right]=[{\stackrel{\&OverBar;}{x}}_U\left(\mathrm{\&alpha;}\right)-Z_{1-\frac{\mathrm{\&alpha;}}{2}}(\mathrm{\&sigma;}/\sqrt{n}),{\stackrel{\&OverBar;}{x}}_U\left(\mathrm{\&alpha;}\right)+Z_{1-\frac{\mathrm{\&alpha;}}{2}}(\mathrm{\&sigma;}/\sqrt{n})]$

Here: ${\stackrel{\&OverBar;}{x}}_L\left(\mathrm{\&alpha;}\right)=a_1+({\stackrel{\&OverBar;}{x}}_o-a_1)\mathrm{\&alpha;},{\stackrel{\&OverBar;}{x}}_U\left(\mathrm{\&alpha;}\right)=a_3-(a_3-{\stackrel{-}{x}}_o)\mathrm{\&alpha;}$

2), calculate owing to supposing the fuzzy of generation:

${\mathrm{\&mu;}}_o^{*}=T(a_1^{\&prime;},{\mathrm{\&theta;}}_o,a_3^{\&prime;})=[a_1^{\&prime;}+({\mathrm{\&theta;}}_o-a_1^{\&prime;})\mathrm{\&alpha;},a_3^{\&prime;}-(a_3^{\&prime;}-{\mathrm{\&theta;}}_o)\mathrm{\&alpha;}]$

3), calculate Fuzzy Test statistic:

${\tilde{Z}}_1\left[\mathrm{\&alpha;}\right]=[\frac{{\stackrel{\&OverBar;}{x}}_L\left(\mathrm{\&alpha;}\right)-Z_{1-\frac{\mathrm{\&alpha;}}{2}}(\mathrm{\&sigma;}/\sqrt{n})-a_3^{\&prime;}+(a_3^{\&prime;}-{\mathrm{\&theta;}}_o)\mathrm{\&alpha;}}{\mathrm{\&sigma;}/\sqrt{n}},\frac{{\stackrel{\&OverBar;}{x}}_L\left(\mathrm{\&alpha;}\right)+Z_{1-\frac{\mathrm{\&alpha;}}{2}}(\mathrm{\&sigma;}/\sqrt{n})-a_1^{\&prime;}-({\mathrm{\&theta;}}_o-a_1^{\&prime;})\mathrm{\&alpha;}}{\mathrm{\&sigma;}/\sqrt{n}}]$

${\tilde{Z}}_2\left[\mathrm{\&alpha;}\right]=[\frac{{\stackrel{\&OverBar;}{x}}_U\left(\mathrm{\&alpha;}\right)-Z_{1-\frac{\mathrm{\&alpha;}}{2}}(\mathrm{\&sigma;}/\sqrt{n})-a_3^{\&prime;}+(a_3^{\&prime;}-{\mathrm{\&theta;}}_o)\mathrm{\&alpha;}}{\mathrm{\&sigma;}/\sqrt{n}},\frac{{\stackrel{\&OverBar;}{x}}_U\left(\mathrm{\&alpha;}\right)+Z_{1-\frac{\mathrm{\&alpha;}}{2}}(\mathrm{\&sigma;}/\sqrt{n})-a_1^{\&prime;}-({\mathrm{\&theta;}}_o-a_1^{\&prime;})\mathrm{\&alpha;}}{\mathrm{\&sigma;}/\sqrt{n}}]$

4), draw subordinate function,

5), calculate:

① $\tilde{Z}\left[\mathrm{\&alpha;}\right]={\tilde{Z}}_1\left[\mathrm{\&alpha;}\right]\&cup;{\tilde{Z}}_2\left[\mathrm{\&alpha;}\right],$ In the time that ambiguity reaches maximum

② $\tilde{Z}\left[\mathrm{\&alpha;}\right]={\tilde{Z}}_1\left[\mathrm{\&alpha;}\right]\&cap;{\tilde{Z}}_2\left[\mathrm{\&alpha;}\right],$ When ambiguity reaches hour

3. work as H _orefusal value minimum (for right-sided test)

4. work as H _orefusal value maximum (for right-sided test)

6), suppose and the subordinate function of I type error is Z _β[α]=[h ₁(α), h ₂(α)]

7), calculate

$Z\left[\mathrm{\&alpha;}\right]>Z_{\mathrm{\&beta;}}\left[\mathrm{\&alpha;}\right]=\frac{k_2\left(\mathrm{\&alpha;}\right)-h_1\left(\mathrm{\&alpha;}\right)}{(h_2\left(\mathrm{\&alpha;}\right)-h_1\left(\mathrm{\&alpha;}\right))+(k_2\left(\mathrm{\&alpha;}\right)-k_1\left(\mathrm{\&alpha;}\right))}$

$\tilde{Z}>{\tilde{Z}}_{\mathrm{\&beta;}}=\frac{\underset{\mathrm{\&alpha;}=0}{\overset{\mathrm{\&alpha;}<1}{\mathrm{\&Sigma;}}}\left((Z\left[\mathrm{\&alpha;}\right]>Z_{\mathrm{\&beta;}}\left[\mathrm{\&alpha;}\right]\&CenterDot;(k_2\left(\mathrm{\&alpha;}\right)-k_1\left(\mathrm{\&alpha;}\right))\&CenterDot;(h_2\left(\mathrm{\&alpha;}\right)-h_1\left(\mathrm{\&alpha;}\right)))\right)}{\underset{\mathrm{\&alpha;}=0}{\overset{\mathrm{\&alpha;}<1}{\mathrm{\&Sigma;}}}((k_2\left(\mathrm{\&alpha;}\right)-k_1\left(\mathrm{\&alpha;}\right))\&CenterDot;(h_2\left(\mathrm{\&alpha;}\right)-h_1\left(\mathrm{\&alpha;}\right)))}$

Here, for any alpha levels, $k_1\left(\mathrm{\&alpha;}\right)=\underset{x}{\min }Z\left[\mathrm{\&alpha;}\right],k_2\left(\mathrm{\&alpha;}\right)=\underset{x}{\max }Z\left[\mathrm{\&alpha;}\right],h_1\left(\mathrm{\&alpha;}\right)=\underset{x}{\min }{Z}_{\mathrm{\&beta;}}\left[\mathrm{\&alpha;}\right],$ $h_2\left(\mathrm{\&alpha;}\right)=\underset{x}{\max }{Z}_{\mathrm{\&beta;}}\left[\mathrm{\&alpha;}\right],$

8) if refusal hypothesis,

There are following 4 kinds of results in this algorithm:

1. work as time, result is the same with the algorithm that M.Arefi and S.M.Taheri proposes, and exists maximum blur level;

2. work as time, result exists least confusion degree; If know be symmetrical, this result is 1. identical with situation; Otherwise this result depends on selected radar type;

3. work as time, that draw is refusal H _ominimum value;

4. work as time, that draw is refusal H _omaximal value.